The Prediction of Earthquake Ground Motions by Regression Model

The prediction of earthquake ground motion is the first priority in the seismic design of a building. This investigation is aimed at proposing regression models for average peak horizontal ground acceleration of earthquakes recorded by seismometers installed at a station in Chiang Mai, Thailand. The majority of earthquakes measured in Chiang Mai occur in seven areas; namely, the regions around Sumatra, Nicobar Island, the Andaman Sea, Myanmar, Laos, China’s western region, and China’s southern region. The earthquakes’ epicenters range from about 10 to 2600 km away from Chiang Mai. The proposed model used 73 earthquakes recorded from 2006 to 2012 and was subdivided according to the magnitudes of the events and earthquake source zones. It was found that the average peak horizontal ground acceleration by regression models was attenuated by the distance from the epicenters. The results of the regression model were in agreement with the records of seven recent earthquakes obtained from Chiang Mai’s seismic station. The regression model has been used in the design of buildings.


Introduction
Southeast Asia was shaped as a result of the interaction between the Indo-Australian, Eurasian, Philippine, and West Pacific tectonic plates. Thailand located inside the Eurasian plate, is delimited by the Andaman thrust within the west, the Sunda Arc within the south, and also the Philippine trench within the east [1]. As a result of recent awareness of earthquake hazards, the Thai Meteorological Department has established quite 20 seismometer stations Thailand. Figure 1 shows samples of ground motion records at a Chiang Mai station (in northern Thailand) from an earthquake event in the regions around Chiang Rai (M 5.9). The distance from the epicenter of the earthquake is about 128 km. One will observe a small peak ground acceleration (PGA) (≈2.29 × 10 −3 g) in the vertical component, and a peak horizontal ground acceleration in the horizontal components (E-W component ≈ 1.63 × 10 −3 g and N-S component ≈ 1.19 × 10 −3 g). The researcher calculated average peak horizontal ground acceleration (PHA avg ) ≈ 1.04 × 10 −3 g.
The attenuation relationship for PHA avg for the Chiang Mai station in Thailand has been proposed in this investigation. The attenuation relationship for PHA avg was determined based on the multiple linear regression (MLR) models by using 132 components of strong motion data from 66 earthquake events. By using the MLR models, the researcher constructed site-specific attenuation relation based on a previously proposed attenuation relation [2][3]. The specific site in this investigation is a Chiang Mai seismic station (CMMT) station located on the rock site.

Characteristics of ground motion in Thailand
Because Thailand is not a country prone to earthquakes, most of the earthquakes felt in Thailand occur outside the country, in places such as Sumatra, Nicobar Island, the Andaman Sea, Myanmar, Laos, and in the west and south areas of China. However, inside the country, Thailand has active faults, which can cause small-to-moderate earthquakes in areas such as Chiang Mai.
Pairojn and Wasinrat [4] presented the observed earthquakes recorded by the Chiang Mai station in northern Thailand from 07/10/2006 to 11/11/2012, as shown in Figure 2.

Site classification
In this study, the researcher chose to investigate data from shear wave velocity (Vs) recorded at the Chiang Mai seismic station (CMMT) in northern Thailand, located on a rocky site

Ground motion prediction
Recently, several ground-motion prediction equations (GMPEs) have been developed. Douglas [6] summarizes all empirical ground-motion prediction equations (GMPEs) used to estimate earthquake peak ground acceleration (PGA) and elastic response spectral ordinates published between 1964 and 2017. Most empirical ground motion attenuation relations are derived from numerical analyses.
For Thailand, Idriss's model [2][3] was selected because it has an appropriate attenuation relation. Idriss suggested that the attenuation relation for motions in western North America [2][3].
The available data included 572 individual horizontal components in rock sites that were used to derive this attenuation model. This study focuses on M ≤ 6.0 and M > 6.0 using local M L and surface wave magnitude M S scales, respectively. The range of applicability is 1-100 km for distance and 4.6-7.4 for M [2][3]. The peak ground acceleration (Y) at rock sites was derived as following equation:

Concept of multiple linear regression models
Regression analysis is a conceptually simple method for investigating functional relationships among variables. The relationship is explained by an equation or a model combining the response (Y) and predictor variables (X 1 , X 2 , …, X p ) [7]. This relationship can be derived from the regression model.
Where ε is assumed to be a random error representing the deviation in the approximation and p is the number of predictor variables. The function f ( X 1 , X 2 , … , X p ) can be classified into two types: linear and non-linear. An example of a linear function is: while an example of a non-linear function is: Note that the term linear here does not describe the relationship between Y and X. This model is linear because in each case the parameters enter linearly, although the relationship between Y and X is non-linear. This can be transformed as follows: where in the equation we have X 1 = In X. The variables here are transformed [7].
A regression model containing only one predictor variable is called a simple linear regression.
The model containing more than one predictor variable is called a multiple linear regression (MLR). The multiple linear regression model is: where β 0 , β 1 , β 2 , … , β p called the regression parameters or coefficients are unknown constants to be estimated from the data. The purpose of the analysis is to estimate the regression parameters or to fit the model to the collected data using the chosen estimation method. The most commonly used method of estimation is called the least squares method (LSE). The LSE method is a procedure to minimize the sum of a squared residual, the difference between an observed value, and the fitted value provided by a model. The LSE method assumes that the error terms are normally distributed, vary constantly, and are independent of each other. The estimated MLR model becomes: The value of Y ̂ is called the predicted value or fitted value. The estimated regression param- To estimate the MLR models, many types of software are used, for example, R, SPSS, and so on.

The multiple linear regression for predicting earthquake ground motion
Pairojn and Wasinrat [4] proposed the MLR model to predict average peak horizontal ground acceleration, PHA avg ( Y ̂ ), in Thailand by including magnitude, and distance as predictor variables. The model was conducted from where M represents the magnitude and R represents the distance. The natural log (ln) of Y and R are used to transform the linear relationship between Y and R, and lnY and M. The value of R 2 = 0.8438 indicated that the earthquake ground motion is predicted by magnitude and distance equal to 84.38%. The results showed that the MLR model is able to predict the average of PHA in Thailand to a greater degree than Idriss's model (Figure 4).

The practicality of ground motion prediction
The MLR model was used on seven new earthquake events that occurred from 05/05/2014 to 05/12/2014. The data are shown in Table 1.
The root mean square error (RMSE) is used for comparing the predicted PHA values from MLR model to the observed PHA values. The RMSE measures the error between a predicted value and an observed value, defined as: where Y ̂ i is the predicted values from MLR model and Y i is the observed values from n observations. The RMSE value is 0.00009.

Conclusions
The MLR model is presented for developing a previous attenuation relationship based on observations. The MLR model is suitable for measuring the attenuation relationship for Thailand because Thailand has few instances of motion data, with most peak ground acceleration being measured at less than 0.1 g. It is expected that this method can be applied to observation sites throughout the country. The MLR model has been used for probabilistic hazard analysis, risk analysis, building design analysis, and many other fields, that is, and construction of nuclear power plants, dams, bridges, and high-rise buildings.